ASTRONOMY AND ASTROPHYSICS. Flare hard X-rays from neutral beams. Marian Karlický 1, John C. Brown 2,3, Andrew J. Conway 2, and Gail Penny 2

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1 Astron. Astrophys. 353, (2) ASTRONOMY AND ASTROPHYSICS Flare hard X-rays from neutral beams Marian Karlický, John C. Brown 2,3, Andrew J. Conway 2, and Gail Penny 2 Astronomical Institute, Academy of Sciences of the Czech Republic, Ondřejov, Czech Republic 2 Department of Physics and Astronomy, University of Glasgow, Glasgow, G2 8QQ, UK 3 Astronomical Institute Anton Pannekoek, University of Amsterdam, Kruislaan 45, Amsterdam, The Netherlands Received May 999 / Accepted 8 November 999 Abstract. A new mechanism is presented for the production of bremsstrahlung radiation from neutral beams (p +,e ) and its possible relevance to flare heating and production of hard x- ray bursts is discussed. Beam electrons lag behind the protons, due to differential drag in collisions with the background, but their longitudinal velocities are closely tied to the protons by the electric field generated. However, collisions with the background also scatter the beam electrons resulting in rms (quasithermal) transverse velocities well in excess of the proton speed. We demonstrate the initial development of this effect using an electrostatic particle simulation with scaled collision rate and then study its full development using an approximate analytic treatment. In particular, the heating of the beam electrons under the bombardment effect of the background is limited by the warm target effect but mean electron energies ( temperatures ) of up to E e.2e p result during the propagation of a neutral beam of initial proton energy E p. Thus, for example, HXR bremsstrahlung in the range 2 2 kev can be generated by protons in the range MeV-MeV. The energy efficiency of the bremsstrahlung production is also limited by the warm target effect but, depending on the HXR spectrum, can exceed.2 of the efficiency of the standard thick target electron beam model. This suggests that the MeV neutral beam model is, in terms of power requirements, unlikely to be the source of HXR-rich flare bursts but that neutral beams able to provide the impulsive flare heating will yield easily detectable HXR burst signatures. Also, while the neutral beam model needs more power ( 5) than an electron beam to yield a given HXR burst flare, it requires a much smaller beam number flux (.7). The issue of the HXR spectral distribution expected from the neutral beam model is also discussed. Key words: acceleration of particles Sun: flares Sun: X-rays, gamma rays Sun: radio radiation. Introduction The association of energetic particle signatures (radio, HXR and γ-ray) with the impulsive phase of solar flares has resulted in Send offprint requests to: M. Karlický (karlicky@sunkl.asu.cas.cz) decades of discussion of whether or not non-thermal particle beams play a role in flare energy transport. In particular, the fact that if HXR bursts are interpreted in terms of the collisional thick target electron beam model (Brown 97), the inferred beam power is sometimes of the same order as the flare power has led to much work on electron beam models of flare heating. Though this model is attractive and receives wide support and use, there have been arguments against it (see e.g. review by Simnett 995). These include for example the fact that some flares have no, or weak, HXR signatures or exhibit no tight correlation between the light curves of HXRs and various thermal signatures (Feldman et al. 98). In addition, the model raises several theoretical concerns. First, how likely are acceleration processes to result in preferential deposition of energy in electrons rather than ions (Simnett 995). Second, how likely is it that a large fraction of magnetic energy release results in acceleration rather than heating? a question that applies to any non-thermal beam model for flare energy transport. Thirdly, in the electron beam model, the beam number flux is also very large. This fact means that the accelerator must be capable of acting on a large fraction of the particle population in a typical coronal loop. Also, as the beam propagates, its large current electro-dynamically generates an electric field which drives a neutralising return current in the background plasma. This is sometimes misinterpreted as having to invoke a return current to get around a problem with the model whereas it is an inevitable physical consequence of the propagation of any beam involving charged particles, as we discuss further below. Because of these objections to the electron beam model, some authors (e.g. Simnett 995, Simnett & Haines 99) have proposed that flare energy transport may instead (or also) involve proton or neutral (p +,e ) beams, and suggested that HXRs are a by-product of these. Various observations have been proposed as favouring proton or neutral beams though the inferences are still somewhat ambiguous. These include Hα impact polarisation (cf. Henoux et al. 99, Fletcher & Brown 995, 998) and γ-ray lines produced by few-mev ions (cf. Ramaty et al. 995, Emslie et al. 997). One of the key issues for such beam models involving protons is how the flare HXR burst is produced, and this is the main issue we address here. Heristchi (986, 987) pointed out that HXRs of (say) 5 kev can be produced by a proton beam of MeV as the pro-

2 73 M. Karlický et al.: Flare hard X-rays from neutral beams tons collide with ambient electrons (p-e bremsstrahlung which is equivalent to e-p bremsstrahlung in the proton frame). This can produce a typical HXR burst with a beam power equal to that required in an electron beam model but with a beam flux µ = m e /m p /2 times smaller. The model fails, however, because the very high proton energies needed would result in γ-ray flare fluxes far in excess of those observed (Emslie & Brown 985). Models invoking heating by protons or neutral beams of lower particle energy (typically MeV) have to date offered no quantitative model of how >5 kev HXRs would result. Before we discuss that further, some further remarks on return currents and neutral beams are in order. When protons are accelerated directionally the current they carry must also result electrodynamically in a neutralising return current if the beam is to propagate. If a proton beam is injected into a dense background plasma, the resulting electric field will result in a neutralising drift of (negative) plasma electrons along the beam direction. This drift current can be dense and slow, compared to the beam velocity, but excited at the beam head through a rapidly increasing volume of the plasma as the beam propagates. In the case of the sun, however, a proton beam is not injected from a gun but is accelerated inside the plasma itself. In this acceleration region the accelerated protons have the same density as the local electrons and the return current may take the form of all the electrons moving at the same longitudinal speed of the protons, the electrons being electrostatically dragged by the protons as soon as they separate by a Debye length. Because the densities are the same, the proton acceleration will essentially result in a neutral beam of proton electron pairs emerging from the acceleration region, essentially as in the models of Martens (988) and Litvinenko & Somov (995). As we will see in Sect. 5, the initial stages of neutral beam production and propagation are crucial in the viability of the HXR production process we discuss. The problem of how such a neutral beam propagates through a background plasma or gas has been discussed in Simnett & Haines (99) and by Brown et al. (998a) in relation to how HXR bursts might result. The key effect is that collisions with the background produce a larger deceleration on the beam electrons than on the protons, causing the electrons to lag behind and producing an electric field. Under certain conditions, which we discuss more fully in Sect. 5, and certainly in the case of a mostly unionised background plasma, or of a background plasma that is not much denser than the beam, the beam charge separation and associated electric field persist and the beam electrons continue to be dragged by the protons. Simnett & Haines (99) have pointed out that this electric field is large enough to accelerate background electrons to high enough energies for HXR production. However, Brown et al. (998a) conclude that the runaway electron flux can only be very small compared to the beam flux since, if it were not, it would very rapidly neutralise the electric field creating it and quench itself. Brown et al. (998a) conclude that this runaway mechanism is completely inadequate to yield electron beam fluxes sufficient for HXR burst production. In this paper we consider an effect, hitherto neglected, in the neutral beam propagation problem, which results in HXR production. Though, as we shall see, it does not suffice to solve the flare HXR problem; it is a process which does result in significant HXR production, and is therefore of interest as a flare neutral beam diagnostic and also in more general astrophysical situations involving neutral beams. In the Simnett & Haines (99) and Brown et al. (998a) analyses, only D motion was considered in the collisional energy transfer process. Brown et al. concluded that, in this case, beam electron motion only comprises longitudinal oscillations, about the beam protons, of very small amplitude and speed. But in reality collisions with the background also scatter the beam electrons, converting beam longitudinal energy into random transverse motion which is not affected by the charge separation field. As the beam propagates we expect the magnitude of this collisionally produced transverse electron energy to grow. In Sect. 2, we present particle simulations which indeed show the initial development of this process. In Sect. 3 we evaluate its full development using an analytic approximation, and in Sect. 4 we discuss the resulting bremsstrahlung radiation signature, and its relevance to flare HXR bursts. In Sect. 5 we discuss the effect of free electrons in the background plasma. In Sects. 3 and 4, however, we assume background conditions are such that they do not much affect the charge separation electric field. For simplicity, however, we treat collisions using the rates appropriate to an ionised plasma, those for a unionised gas differing only by factors of 2 3, within the orders of magnitude effects with which we are concerned here. We emphasise that this is the first treatment of this particular effect in neutral beams and is aimed at illustrating its potential importance in flare HXR production, so the analysis is deliberately kept as simple as possible. In particular the following assumptions /idealisations are made: (a) by a beam we mean, in common with previous flare beam modelling, a well collimated particle distribution i.e one which has v v ; (b) the beam is initially cold i.e. v 2 v 2 v 2 though as it propagates the electron component becomes hot ; (c) the beam is very tenuous compared to the background gas density; (d) the beam kinetic energy density is larger than the thermal energy of the background gas so the latter is significantly heated by it but because of (c) never to a high enough temperature to produce HXRs or to modify our analysis of beam collisions; (e) the mean free path of beam particles is long compared to all relevant gradient lengths and the beam duration is long compared with the particle stopping time. Consequently the interaction of beam and gas is treated kinetically and as a quasi-steady spatially extended structure, rather then a fluid boundary interaction (shock) problem; (f) the interaction is described purely in terms of mean particles with no coherent effects considered; (g) for simplicity we consider the problem basically in terms of monoenergetic injection spectra discussing other spectra merely by summing contributions from different energies. These last assumptions (f) and (g) are those which most need further investigation. In a purely neutral background gas they are quite reasonable but in the ionised coronal region of initial beam propagation it is possible that coherent generation of plasma waves will modify the beam behaviour from our purely

3 M. Karlický et al.: Flare hard X-rays from neutral beams 73 collisional description. This is certainly the case for monoenergetic charged beams propagating in an ionised background, as widely discussed for Type III radio burst production (McLean & Labrum 985), and also under some background plasma conditions even for beams with monotonic decreasing injection spectra (Emslie & Smith 984). However both theoretically and even moreso observationally (Melrose 98, Benz 993) the effects in these cases, if any, are mainly to redistribute energy among beam electrons rather than to extract much energy from the beam. For this reason coherent effects have, following Brown (97), commonly been neglected in modelling of HXR production by electron beams. In the present case their importance is even less clear since the beam itself has nett neutrality. We note furthermore that most of the collisional heating of beam electrons, which lies at the heart of our model, occurs well along the beam column density path where the background is substantially neutral. Thus while the issue of coherent effects does need addressing for the neutral beam model we do not address it in the present paper for the neutral beam. 2. Numerical simulations We investigate the propagation of an initially monoenergetic neutral beam. For simplicity particles of the background plasma are assumed immobile so that they provide only beam electron scattering and collisional drag on the beam particles, i.e. as already stated, no beam current neutralization by the background plasma flow is considered. If we assume a neutral beam with an infinite cross-section or a beam propagating along straight magnetic field lines, the electron beam heating can be considered as -D in the electric field and 2-D in electron beam velocities. Thus, to demonstrate the heating, a modified -D electrostatic particle numerical code can be used. Even in the simplest case of an infinite homogeneous neutral beam, this purely temporal problem is complicated by the large range of time-scales involved from the electron plasma period up to the proton collision time. Since the proton collision time is long compared to the electron collision time and extremely long compared to the electron plasma period, in this Sect. we simplify the problem by considering collisional effects on electrons only i.e. we approximate beam protons as infinitely heavy. Moreover, to shorten computation times the energy losses and scattering of electrons are artificially increased. For this reason we are speaking only about a scaled demonstration of the initial stages of the heating process. In the standard numerical electrostatic particle code (Birdsall & Langdon 985) a homogeneous infinite neutral beam cannot be simulated, even with periodic boundary conditions. The reason is that in the code there can be no charge at infinity and it is such charges which can be thought of as the source of the charge separation electric field. To overcome this, we made a simulation using a code with periodic boundary conditions, but with the neutral beam shorter than the length of the system so that beam-end effects are present. For the length of the system we took L =2π (N G = 256 grid points) with the neutral beam initially between L/4 and 3L/4, i.e. occupying half of the system length. numerical electrons and numerical protons were considered. It is useful to make particle code computations in the frame of the beam protons, where the initial velocities of both electrons and protons are zero. Nevertheless, for scattering computations the initial beam speed in the background plasma frame of V p = code velocity units (L/(N G t)) needs to be defined (scaled code velocities are denoted by upper case V ). Our numerical scheme then proceeds as follows. At each time step t =.2units ( unit. beam plasma periods), we simulate the effect of collisions by reducing the total speed V TOT = (V p + V X ) 2 + VPER 2 of each electron by V TOT = C/VTOT 2. Here C =.is a scaled up collisional constant, V PER is the speed across the longitudinal (beam) direction and V X is the longitudinal speed relative to the protons. This deterministic reduction in V TOT has the effect of reducing V X for each electron, by an amount determined by its pitch angle α which is also altered by collisions but in a stochastic fashion which we treat fully in accordance with the approach of Bai (982). In subsequent time steps the electrons fall behind the protons creating an electric field (determined in the code by Poisson s equation) which accelerates them, bringing them back to the proton speed. The net result is a transfer of longitudinal proton energy into random perpendicular and longitudinal velocity components of the electrons. The simulation described above was used in two regimes: a) with and b) without electron pitch angle scattering, energy losses due to collisions being considered in both cases. First, let us consider the case with scattering (ignored in Brown et al. 998a). The computations were made up to 23 plasma periods of the beam electrons. Oscillations of the beam electron component, caused by collisional drag and the resulting electric field, are seen in Fig., where the mean parallel component velocities (in the proton beam frame) are shown for both electrons (dashed line) and protons (thick line). In the bottom part of this figure there is an enlargement of the electron oscillations. The state of the electron velocity distribution at t = 23 plasma periods are shown in Fig. 2, as velocity components and the cosine µ = cos α of electron pitch angles. The velocities of electrons are shown in units where the initial proton speed is. All initial perpendicular velocities were taken to be zero. The protons having constant speed stay inside the grid, whereas electrons are kept there by the charge separation electric field. A small number of electrons escape for numerical reasons, but this effect is not important here. The most important result is that the electrons are heated, in that energy is accumulated in their perpendicular velocity distribution see also Fig. 3, where the evolution of the electron distribution function in electron energy (VTOT 2 ) is shown. To illustrate the key role of electron pitch angle scattering, computation of the case without scattering was also carried out (Fig. 4) and we see that the gain in energy is small, consistent with previous results presented in Brown et al. (998a). From these scaled numerical simulations, we estimate that at the very beginning of this electron beam heating the heating rate

4 732 M. Karlický et al.: Flare hard X-rays from neutral beams Fig.. Time evolution of mean parallel-component velocities of electrons (dashed line) and protons (thick line). Velocities are expressed in the the proton beam frame. In the bottom part of the figure a detail of the electron component oscillation for plasma periods is shown. is about the initial electron beam energy per electron collisional stopping time. To obtain the total heating gain of beam electrons the proton collisional deceleration needs to be included. If this heating rate continued the electrons would attain energies comparable to E p as the protons stopped but in reality, discussed in the analytic treatment below, the heating rate of the electrons declines as they heat. However, such an extension of the computation is not possible in this type of numerical model because of beam end effects. To take the calculation further, having demonstrated the basic heating effect, we now revert to an approximate analytic treatment. Fig. 2. Electron distribution at t = 23 plasma periods for the case with scattering. VTOT is the total electron velocity in the background plasma frame, VPER is the perpendicular velocity, VX is the parallel velocity in the proton beam frame, MU is the cosine of the electron pitch angle. The initial beam velocity in the background plasma frame is units. 3. Analytic treatment For the moment, as in the numerical simulations, we consider for simplicity only the case of an injected beam which is monoenergetic; with speed v p, energy E p = 2 m pv 2 p, and of density n b at injection. (Aspects of the case with an energy distribution will be discussed in Sect. 4.) As the beam protons decelerate by colliding with the background, these values evolve to v p, E p and n p at time t, distance x and atmospheric column density N along the path. The particle flux F = n p v p = F is constant in a steady state (for this mono-energetic case) and corresponds, over injection area A, to a total injected beam power P = AF E p. According to the D analysis of Brown et al. (998a) (end of their Sect. 2), the accompanying electrons on average follow the Fig. 3. Evolution of the electron distribution function during collisional deceleration and scattering. Its state at 45, 9, 35, 8, and 225 plasma periods is shown by the solid, short dashed, dot-dashed, dot-dot-dotdashed, and long dashed lines, respectively. The electron energy equals VTOT*VTOT expressed in the background plasma frame. protons at a tiny distance (due to the differential collisional drag) with the same mean speed, though undergoing small amplitude, high frequency, electrostatic oscillations about their mean position. The spread in the electron speed was not considered in the

5 M. Karlický et al.: Flare hard X-rays from neutral beams 733 Fig. 4. The same as Fig. 2 but without scattering. D approach, as collisions were assumed only to cause a deterministic deceleration of the electrons. In Sect. 2 above, however, we have seen in a 2D treatment where (transverse) scattering, and not just energy loss, is allowed that both the mean value and the variance of the transverse velocity increase as the beam propagates. This is because changes in transverse speeds are not inhibited by the charge separation electric field as is the longitudinal speed. This growth and spread in transverse electron energy can be visualised in the local beam frame as essentially a heating of the beam (target) electrons under the collisional effects of the bombarding atmosphere. The flow of energy here is not immediately obvious. In the beam frame, the initially cold beam electrons are apparently being heated by these collisions. This heating can be seen in the plasma background frame as the beam electrons developing a distribution in velocity whilst following the longitudinal motion of the protons. In the plasma background frame, the electrons and protons are both losing energy to the background particles. However, the electrons are also gaining energy from the protons because of the dragging and, according to the results of Sect. 2, must be doing so faster than they are losing energy to the background plasma. If left alone, these beam electrons would quickly stop, and only heat up very slightly. However, the fact that they are driven, by the electrostatic dragging action of the protons, means that they continue to heat up. We want to estimate the mean random electron kinetic energy achievable by this process during the stopping lifetime of the protons, and also to evaluate the efficiency of the resulting bremsstrahlung radiation to see if it can contribute to the flare HXR production. Though we recognise that the spread in electron energy need not be Maxwellian, it is evident from the numerical results of Sect. 2 that there is a considerable spread skewness about the mean and we will sometimes deal with this in terms of a beam electron temperature. The beam protons lose energy via beam collisions in two ways. The first is via direct collisions with the background particles. This loss is dominantly to background electrons, and occurs at a rate (de p /dt) = Kn v p /µe p where K =2πe 4 Λ, where Λ is the Coulomb logarithm, µ = m e /m p and n is the local background density (cf Emslie 978). Here we have assumed the background to remain cold. The second is via the electrostatic drag exerted on the beam protons by the beam electrons. Energy transfer from the beam protons to the beam electrons in their collisions with the background can be written for a mean beam electron of energy Ēe as (de/dt) 2 = (dēe/dt) = 2n v p f(e p, Ēe)K/µE p where the factor f(e p, Ēe) allows for the decline in collisional energy transfer to the beam electrons, from the background electrons and ions, as the beam electron mean random speed increases and exceeds the speed (v p ) of the bombarding background particles. If the beam electrons are cold at injection (rms speed v p ) then there the cold target limit f =applies and we see that rapid heating should occur at the rate dēe/dt =2n v p K/µE p which means (just as found in our simulations Sect. 2) that the random E e will increase on the time-scale of an electron collision time. This means, however, that in a very small fraction ( µ) of the proton stopping time the electron random speed will exceed the longitudinal velocity of the beam, i.e. of the bombarding background (or Ēe >µe), and the warm target situation (Trubnikov 965) of decreasing f will apply for the remainder of the beam propagation. Since this will in fact apply over almost all the beam length, and since the electron energies in the very initial phase are not relevant to HXR production, we will simplify our analysis by adopting a small beam electron random energy Ēe = µe p at injection and using the warm target f throughout. To do so we adopt a rough analytic fit to the graph of f in Emslie et al. (997), viz. ( ) 4/3 f = f(e p, Ēe) µep () Ē e where we have equated Ēe here with temperature (energy units) in that graph. In this regime the evolution of E p and Ēe are then well described by de p dt dēe dt = (+2f) Kn v p µe p Kn v p µe p (2) =2f Kn v p µe p (3) (It is important to recall here that Ēe /= µe p since the random electron speed is not equal to the beam speed v p ). In (2) and (3) we have retained only the lowest order terms in f: in this domain (covering almost the whole trajectory), the energy equation of the protons is dominated by their direct energy losses to the background i.e. f is assumed small enough to justify 2f but still substantial enough to result in significant energy transfer from the protons. These equations can be solved to give

6 734 M. Karlický et al.: Flare hard X-rays from neutral beams Ep/Epo.6.5 Ee/Ee* N/No Ee/Ee* Ep/Ep Fig. 5. a The ratio of proton energy to initial proton energy Ep/Epo (solid line) and electron energy to maximum (final) electron energy Ee/Ee (dashed line) vs. the ratio of traversed column depth to beam stopping column depth N/No. b The ratio of electron energy to maximum (final) electron energy Ee/Ee vs. the ratio of proton energy to initial proton energy Ep/Ep. E p (N), Ēe(N) and also Ēe(E p ) where N(x) = x n (x )dx is the ambient column density measured from injection. (Note that we neglect a minor correction due to proton pitch angle scattering on the background protons). The solutions are E p (N) =E p ( N N ) /2 (4) where N = µe2 p 2K =2. 2 cm 2 Ep(MeV) 2 (6) is the column density required to stop the protons and the maximum mean electron energy Ēe, attained by the electrons as the protons stop, is given by Ē e =2 3/7 µ 4/7 E p.83 2 E p or (7) [ Ē e (N) =Ēe = Ēe ( Ep E p ) 7/3 ] 3/7 = [ ) ] 7/6 3/7 ( NN (5) Ē e (kev ) 8.3E p (MeV) It follows that collisional effects in the propagation of a neutral beam of particle energy say MeV at injection give rise to random transverse motions of the beam electrons with mean energies up to around 8 kev. (We observe, however, that in

7 (7) the factor 2 3/7 cannot really be distinguished from unity, given the approximations made.) In Fig. 5a we show E p /E p (solid) and Ēe/Ēe (dashed) as functions of depth N/N in the target, and in Fig. 5b Ēe/Ēe is plotted againsts E p /E p.we can see that the electrons closely approach the maximum Ēe rather rapidly so that the hard electrons capable of emitting HXRs emit throughout most of N, the moreso since the beam emissivity is concentrated toward N = N as the beam density increases. The mechanism we have discovered is thus capable of producing, from a MeV neutral beam, electrons of sufficient individual energies for flare HXRB production, but we still have to determine whether the process is efficient enough to produce the observed HXRB bremsstrahlung intensities and, if so, to explore its observational properties. We have also checked this by more exact treatment of the above problem with f(e,t) as described in Emslie et al. (997). Namely, the total electron heating was computed by a numerical integration of the following equation: tstop Pdt = tstop (E 3/2 p C M. Karlický et al.: Flare hard X-rays from neutral beams 735 µct)/3 Gdt, (8) where t stop is the proton stopping time, C=2 3/2 πe 4 Λn/m /2 p, and G = ν(x)/mu ν (x) is the exact warm target expression. (Our ν corresponds to µ in Emslie s notation.) The energy gain of beam electrons was computed for different proton beam energies in the range of MeV. Results are presented in Fig. 6, where the initial electron energies (dashed line) and the resulting mean electron energies after the heating process (solid line) are shown. These results are in very good agreement with the results obtained using empirical fit used above. 4. Bremsstrahlung radiation 4.. Efficiency We define bremsstrahlung efficiency of a beam as the bremsstrahlung power emitted per unit beam power input. As discussed further below the answer depends somewhat on the photon energy (ɛ) range considered, on the spectrum emitted, and on the bremsstrahlung cross-section approximation adopted. Here, however, we are mainly interested in comparing the thick target neutral beam bremsstrahlung efficiency with that of a thick target electron beam. Therefore, so long as we consider essentially the same energy ranges and use the same crosssection in both cases, these issues should be secondary. For simplicity in this comparison we will therefore use the Kramer s bremsstrahlung cross-section: Q(ɛ, E) = Q (9) Eɛ At first sight the neutral beam model might be expected to be efficient in that the electrons are at least quasi-thermal, and pure thermal bremsstrahlung sources can be very efficient. This is, however, only the case for the ideal isolated (iso)thermal source for which radiation is the only energy loss. In real nonisothermal sources there are power losses by conduction and Fig. 6. The total energy of the beam electrons after the heating process (solid line) and their initial energies (dashed line) vs the initial proton beam energy. ξ(τ) / ΕΜ Fig. 7. Emission measure of the beam electrons differential in temperature as a fraction of the total emission measure (ξ(t )/EM) vs. beam electron temperature as a fraction of the maximum (final) temperature (T/T ). escape. In the present case the situation is that the hot electrons only remain so because they are continuously heated by extracting energy from the protons to offset the collisional deceleration they would otherwise undergo in collision with the cold background. One can in fact think of the beam electrons as producing their bremsstrahlung in a cold thick target background but with continuous driving by the protons. As we have seen, due to the warm target effect, however, this transfer of energy from the protons to the electrons is rather inefficient and we may consequently expect a rather high beam power requirement to get.5 T/T*

8 736 M. Karlický et al.: Flare hard X-rays from neutral beams enough into the beam electrons for thick target HXR production i.e. a rather low efficiency. Calculation of the radiation efficiency is complicated by the fact that the actual distribution of the beam random electron energies is unknown. In Sect. 3 we estimated only a mean value Ēe and its evolution, though we also observed on the basis of our simulations that the distribution possibly has a quasi- Maxwellian form. We will therefore proceed by considering the efficiency in the two limiting cases where the local electron distribution is treated as monoenergetic (delta-function at E e = Ēe) and as Maxwellian with T = Ēe Mono-energetic E e (N) Assuming that radiation of beam electrons is due to collisions with background plasma protons, the power in the bremsstrahlung emission, per unit photon energy ɛ, from the entire length of the beam, for neutral injection flux F, can be expressed as n v e L(ɛ) =ɛj(ɛ) =AF Q dt () t(e e>ɛ) E e where the time interval is governed by the proton deceleration, spanning the range of proton energies E p from to that at which E e first exceeds ɛ, namely by (5) ( ) ] 7/3 3/7 ɛ E p = E p [ () E e (Note, in this monoenergetic case E e Ēe.) Thus, writing dt = de p / de p /dt we get, using (4) (7), for the HXR spectral luminosity Ep[ (ɛ/e e ) 7/3 ] 3/7 = AF Q ( µ ) 3/4 P Q = 2 K n v e E e [ (ɛ/ee ) 7/3 ] 3/7 L(ɛ) =ɛj(ɛ) = µe p n v p K de p = x /2 dx [ x 7/3 ], 3/4 ɛ<e e (2) where P = AF E p is the beam power and x = E p /E p. The most natural definition of neutral beam bremsstrahlung efficiency η n in this case is the ratio of the total bremsstrahlung power (all ɛ) top, namely η n = L(ɛ)dɛ = Q µ /2 ( ) /2 Ep P Ee K [ (ɛ/ee ) 7/3 ] 3/7 = 3 ( µ 7 2 E e x /2 dx dɛ (3) [ x 7/3 ] 3/4 ) 3/4 B(9/4, 7/4) Q E e K where B is the beta function. We want to compare this with the efficiency of a cold thick target electron beam. In particular, we will compare it with the efficiency of a monoenergetic electron beam of energy E e at injection since this produces a photon spectrum, like (3) restricted to <ɛ<e e. The result is η e = Ee Ee Q E e AF AF E e ɛ= ɛ E e K de e = Q 2K E e (4) Thus the relative efficiency of the neutral beam is by (3) and (4) η n = 6 ( µ ) 3/4 η e 7 B(9/4, 7/4).2 (5) 2 independent of E. This figure bears out our earlier expectation that η n <η e but in fact the ratio is not very small and makes the neutral beam model of considerable interest. Result (5) says that we need 5 times more power in a monoenergetic neutral beam to produce a prescribed bremsstrahlung flux than we need in an electron beam. On the one hand this suggests that neutral beams are not a serious candidate to explain HXR burst intensities from HXR-rich flares since to explain these with an electron beam already requires a beam power of the order of the total flare power. That is the neutral beam compounds the beam power problem for such events, though it does reduce the beam number flux required by a factor of (E e /E p )(η e /η n ).. On the other hand, result (5) also means that a MeV neutral beam carrying a power of order the flare power (i.e. able to act as a flare energy transport mechanism) will produce an easily detectable HXR burst in the 2 2 kev range. Result (5) must be treated with some caution, however, in that we are not really comparing like with like. All we have done is to compare the photon energy integrated bremsstrahlung from monoenergetic electron and neutral beams. But, apart from having the same upper cut off (ɛ E e ), the bremsstrahlung spectra from those two models are not the same that from a monoenergetic electron beam is L(ɛ) E ɛ ( ɛ E ) which is not the same as () from a monoenergetic neutral beam. To discuss the relative efficiencies of the two models in a reliable way, we really want them to produce the same photon spectrum as well as total luminosity. This is made difficult by two facts that we do not really know the electron energy distribution about Ē e for the case of a monoenergetic neutral beam. Second, it is not clear how to deal with more general neutral beam injection spectra which may be required to match HXR spectral observations cf. Sect To deal with the first problem, we examine the effects of replacing a monoenergetic E e distribution at E e = Ēe with a local Maxwellian at T = Ēe Maxwellian E e (N) The bremsstrahlung spectrum from a thermal source in the Kramer s approximation can be written ( ) /2 8 L(ɛ) = Q πm e T ξ(t ) T /2 e ɛ/t dt (6)

9 where ξ(t ) is the differential emission measure function which in our D case is M. Karlický et al.: Flare hard X-rays from neutral beams 737 ξ(t )= An en dt/dx = An e dt/dn (7) where n e is the beam electron density given by n e = F /v p (N), and n is the background proton density, since in our problem the bremsstrahlung is from collisions of hot beam electrons with background ions (assuming the latter to be denser than the beam). Writing dt/dn =(dt/de p )(de p /dn) and using () (3) we get ξ(t )= 3 ( µ ) 3/7 ( mp ) /2 AF E /2 p K ( ) [ 4/3 ( ) ] 7/3 5/4 T T (8) T T = 9 [ EM t 4/3 t 7/3] 5/4 t 4T where t = T/T and the total emission measure is T ( mp ) /2 PE /2 p EM = ξ(t )dt = µ (9) 2 K Fig. 7 shows ξ(t )/EM which emphasises that the differential emission measure is concentrated near T = T. Inserting (8) in (6) and integrating over ɛ we find the neutral beam radiation efficiency in this case to be η n = L P = P L(ɛ)dɛ = = 3 2 /4 µ 3/4 B(9/4, 7/4) Q 7 π /2 K T (2) so that now, using (4), the relative efficiency is η n = 2 ( µ ) 3/4 B(9/4, 7/4)=.22 (2) η e 7π /2 2 which is essentially the same as when E e was regarded as monoenergetic at Ēe Spectrum Inserting (8) in (6) allows us to compute the spectrum J(ɛ) expected from a monoenergetic neutral beam when E e is taken to have a local Maxwellian distribution of T = Ēe, namely: L(ɛ) = 2µ/2 Q ( π /2 K P t 5/6 t 7/3) 5/4 e ɛ/tt dt (22) The result is shown in Fig. 8 along with the spectra for the monoenergetic case (3) and also of an isothermal source, with the same total emission measure, all at T = T = E e. The thermal and isothermal spectra are very similar since ξ(t ) peaks sharply at T while the monoenergetic spectra cuts off at ɛ = T Fig. 8. Spectrum of hard X-rays under various assumed initial beam distributions: thermal (solid), iso-thermal (dashed), power law (dotted), and monoenergetic (dashed-dotted). Efficiency Ratio δ Fig. 9. Relative efficiency of neutral beam hard X-ray production (compared to an electron beam) vs. power law index δ of initial beam distribution. because no Maxwellian tail exists in this case. The overall behaviour of the thermal case is that J(ɛ) ɛ for ɛ T and e ɛ/t /ɛ for ɛ T. Except over a limited range, none of these spectra resemble observed HXR spectra which are well approximated as power laws. In order to compare accurately the efficiency of the neutral beam with an electron beam we should really do so for sources which yield the same HXR spectrum, and for spectra similar to those observed. The natural way to think of getting a broader spectrum from the neutral beam is to have a range of E p (such as a power law), rather than a single E p at injection. This poses a physical problem if the distributed E p occurs across the area A for then, along each beam path, there is a range of proton speeds and it is not clear how this modifies the electrostatic/charge separation arguments on which our neutral beam model has been based. Electrodynamics does not demand that each proton be accompanied by an electron of the same longitudinal speed

10 738 M. Karlický et al.: Flare hard X-rays from neutral beams but only that the mean longitudinal speed of the beam electrons (or electron current) equals the mean proton speed (current). If in fact the electrons do follow the protons on a one-to-one basis then the resulting HXR spectrum, and the radiation efficiency, can be found simply by integrating our previous L(ɛ) results over a distribution F (E p ), but establishing whether or not this is the case physically is beyond the aims of our present study. However, one situation where this approach is valid is where the neutral beam comprises monoenergetic proton beams along each path in A but with a distribution of E p across A. We therefore here evaluate the spectrum and efficiency in this situation for the particular case where the total beam power injected is distributed in E p over A with a power law form that is, with P (E p ), the beam power per unit E p given by P (E p )=(δ 2) P ( ) δ Ep (23) E E where P is the total power at E p E. If we then replace P in (3) by P (E p )de p from (23) and integrate over E p we obtain (treating E e as locally mono-energetic at Ēe) ( µ ) 3/4 Q L n (ɛ) =(δ 2) 2 K P (24) ɛ/2 3/7 µ 4/7 = 3 7 ( Ep ( µ 2 E ) δ [ (ɛ/ɛ ) 7/3 ] 3/7 x /2 dx de p ( x 7/3 ) 3/4 E ) ( ) 3/4 2 δ 3 B(9/4, 7 (δ /6))Q ɛ P K ɛ where ɛ =2 3/7 µ 4/7 E. Thus a power law flux in E p of index δ yields a power law flux J(ɛ) of index δ just as for a thick target electron beam for which the result is explicitly L e (ɛ) = ( ) 2 δ Q ɛ P (25) δ K ɛ where P is the electron beam flux above ɛ. To yield exactly the same flux and spectrum, i.e. L n (ɛ) =L e (ɛ) for all ɛ this implies a relative efficiency η n = P = 3 ( µ ) 3/4 3 (δ )B(9/4, (δ /6)) (26) η e P which is shown in Fig. 9 (solid curve) as a function of δ. We see that the relative efficiency is of the same order as found for the monoenergetic case for small δ but increases somewhat for larger δ. We note, however, that we are again not quite comparing like with like here. The electron power law cut-off below E e = ɛ yields a photon spectrum J(ɛ) γɛ γ with γ = δ at ɛ ɛ and γ =at ɛ<ɛ. The neutral beam power law cutoff at E = ɛ /2 3/7 µ 4/7 yields exactly the same spectrum (with γ = δ ) for ɛ ɛ but for ɛ<ɛ produces a photon spectrum of the same form (3) as a monoenergetic neutral beam which is not J(ɛ) ɛ but rather falls below the spectrum emitted by the electron beam case. We can also compare the electron beam with the neutral beam where allowance is made for the local spread in the neutral r Fig.. Comparisons of the HXR spectra resulting from a power-law neutral beam injection spectrum, with a locally Maxwellian beam electron distribution. The curves are plots of L(E )/L() from (29) as functions of r = ɛ/ɛ for spectral indices δ =4, 5, 6, 7. beam E e by assuming this distribution to be Maxwellian. That is, we insert (23) in (6) and (7) and obtain L n (ɛ) = 27 ( µ ) ( ) 3/4 2 δ Q ɛ 4π /2 2 K ɛ P Γ(δ )B(9/4, 3 (δ /6)) (27) 7 so that here, comparing with the electron beam (25) we get η n = P 6 ( µ ) 3/4 η e P 7π /2 2 Γ(9/4)Γ(δ)Γ( 3 7 δ /4)Γ(3 δ +4/7) (28) 7 which increases with δ faster than (26) by the factor Γ(δ)/(δ )=Γ(δ ) and η n /η e from (5) becomes for large δ. Here, however, the problem of comparing like with like becomes very serious because (27) is only a power law above ɛ = ɛ if P (E p ) isapowerlawdowntoe p for which the total beam power is infinite. This is also true of the electron beam if one requires J(ɛ) ɛ γ for all ɛ and we are then comparing two infinite quantities. Since J(ɛ) is measured only over a finite interval and with finite accuracy, we can usefully ask how much L n (ɛ) differs (e.g. at ɛ ɛ ) between power laws P (E p ) with and without a cut-off i.e. cut-off at E p = E and at E p =. The relevant expression is L n (E ) L n () = Γ( 3 7 δ +4/7) Γ(δ 2)Γ( 3 (29) 7δ /4)Γ(9/4) ɛ/ɛ u 3/7 y δ 3 e y u 3δ/7 5/4 ( u) 5/4 dydu which is shown in Fig. as a function of r = ɛ/ɛ for δ = 4, 5, 6, 7. We see in fact the spectra differ enormously for r<, i.e. for photon energies below the final electron energies T (E )

11 M. Karlický et al.: Flare hard X-rays from neutral beams 739 corresponding to the cutoff in proton energy E. The difference is more pronounced for larger δ, and is only small when r. Thus the efficiency comparison (28) is not really meaningful in terms of comparing electron and neutral beams producing similar photon spectra, other than in the unphysical limit of E p = E. In summary, what these results show is that the exact bremsstrahlung spectrum produced by a neutral beam is rather sensitive to the spread in the local electron energies E e along the beam path and that the actual efficiency will need to be computed using more elaborate treatments of the distribution functions (including electrodynamics). However, from the cases where we have managed to compare the efficiencies of the neutral and electron beams in a convincing manner in our approximations to the E e distribution, the efficiency of neutral beams seems to be of order 2% of the electron beam efficiency. 5. Effect of plasma electrons Central to the above mechanism for HXR production is the assumption that the neutral beam survives as such along the stopping length of the protons without the beam electrons being stripped from them. As discussed by Brown et al. (998a) and by Simnett & Haines (99) the presence in the background gas of free electrons, which we have ignored here so far, modifies the electrodynamics and can result in electron stripping. Since there will always be some free background electrons present in the real solar atmosphere, even in the low chromosphere, we reconsider here whether a neutral beam can in fact survive the propagation process. In doing so we correct an error in the discussion of this problem by Brown et al. (998a) which may change their conclusions under some conditions. When a neutral beam enters a region with free background electrons these respond to the p-e charge separation electric field and so contribute to the return current. Denoting by j e,j p, and j o the current densities of the beam electrons, beam protons, and ambient electrons respectively, steady current neutralisation then requires j e + j o = j p. Background electron motion results in j o > so that j e < j p and since n e = n p in the beam j = nev implies v e <v p so that the electrons fall progressively further behind the protons. Whether this effect is large enough to affect our conclusions regarding HXR production depends on how large a current j o the background can carry in response to the electric field and this will depend on the free electron density and on the effective collisionality (resistivity) of the background plasma. The dynamics of this process are described by Eqs. (6) (8) of Brown et al. (998a) which, in the above notation (dropping subscript e for simplicity and denoting the electric field by E) are dv m e = ee C(v) (3) dt dv o m e dt = ee C(v o) (3) de dt =4πe(n ov o n v) (32) where v = v p v (33) and C(v) is the collisonal drag force. If the beam persists long enough, the solution of these equations approaches the steady state (Brown et al. 998a) v = n o n v o (34) with v o determined by C(v o )=C(v). (35) The error in Brown et al. (998a) was the argument that (35) implies v = v o (meaning that the beam electrons are stripped from the protons and join the plasma electron drift) since C(v) is monotonic. In Brown et al. the treatment was solely -D and deterministic and the beam and plasma were cold so no spread in v, v o were included. In real plasmas C(v) is, however, only monotonic ( /v 2 ) for drift speeds well above the thermal speed v t but turns over at around the thermal speed and tends to v/v 3 t for slow drifts. Unless the plasma is very highly conductive (i.e. hot) the plasma drift speed v o, and hence the plasma drift current j o = n o ev o, at which (33) reaches a steady state are smaller than (v o = v) found by Brown et al. In short, the plasma electrons are more collisional than the beam electrons and their contribution to neutralising j p will be less likely to result in beam stripping than suggested by Brown et al. To quantify this for general drift speed v d we adopt (with v t =(T/m e ) /2 at temperature T ) C(v) = 4πΛe4 n o v d m e (vd 2 + (36) v2 t ) 3/2 which for v d = v o v t is C(v) 4πΛm/2 e e 4 n o v o (37) (T o ) 3/2 where T o is the background plasma temperature. (Here we have essentially taken the Spitzer conductivity limit for a fully ionised gas and ignored any anomalous effects these classical are pessimistic in minimising the collisionality and hence maximising j o ). For the beam electrons (v d = v v t )itis C(v) 4πΛe4 n o m e v 2 f (38) where we have added the factor f from Eq. () to allow as before for the fact that, as they propagate, beam electrons become warm relative to the bombarding background. Inserting (37) and (38) in (35) to find v o we get v o = (T o/m e ) 3/2 v 2 f (39) and inserting this in (34) gives v v p = f n o n (T o ) 3/2 m 3/2 e v p v 2 (4)

12 74 M. Karlický et al.: Flare hard X-rays from neutral beams Once the beam electron heating, which we have described in previous Sections, is well developed we approach v 2 v p /µ 3/7 giving v 3 4 f n [ o To (K)/ 7 ] 3/2 (4) v p n E p (MeV) with f. Thus v v p and stripping is negligible even for quite high beam densities even moreso in the chromosphere than in the corona since n o To 3/2 falls across the transition zone At the start of beam propagation, however, we have v v p,f and so v v p = n o n (T o ) 3/2 m 3/2 e vp 3.7 n o n [ To (K)/ 7 ] 3/2 (42) E p (MeV) This suggests that if the above steady state is approached and the beam starts and in the corona then very substantial electron stripping may occur ( v v p ) even for an E p of MeV unless: T o 7 K, which is unlikely; and/or the beam density is very high (n n o i.e. essentially all of the plasma is accelerated; and/or substantial anomalous resistivity (collisionality) exists. If on the other hand the beam started in the chromosphere falls by a factor across the transition zone, that the number density of free electrons drops with depth, and that the beam becomes denser as the protons slow down, all make survival of the neutral beam more likely. This analysis rests on consideration of the steady state. This is approached on a timescale τ m e v p /C(v). At the start of the beam the relevant numbers yield the facts that nt 3/2 o τ = m/2 e 4πe 4 Λn o [ 2me E p m p ] 3/2.sec [E p(mev)] 3/2 n o / cm 3 (43) which is short compared to the time for the beam to reach the chromosphere or to stop unless E p MeV and n o 9 cm Discussion We have shown that substantial HXR signatures can arise from neutral beams by heating of the electrostatically dragged beam electrons in collision with a neutral background. This process may be important in a variety of astrophysical situations involving fast neutral streams. The process can also yield easily detectable HXR signatures from solar flares for neutral beams of sufficient power to produce flare heating, provided the neutral beam can survive propagation among the free electrons of the solar flare background plasma. We have corrected an error in the previous analysis by Brown et al. (998a) of neutral beam propagation and shown that neutral beams can survive, once developed, in the flare corona and can survive from their outset in chromospheric conditions. However, both theoretical and observational considerations favour a coronal origin for accelerated beams. Theoretically reconnection and associated acceleration mechanisms are generally thought to operate best in coronal conditions. Observationally, the observed synchronism of HXR footpoints within.s (Sakao 994) cannot be explained by neutral beams originating at one chromospheric footpoint and travelling to the other, since MeV protons take of order s to do so. These data rather favour symmetrical downward propagation of beams from a common central looptop source, as does the time of flight analysis by Aschwanden & Schwartz (995) and by Brown et al. (998b). The issue of whether magnetic and other conditions can exist whereby a neutral beam can be produced and survive under coronal conditions requires further investigation and is intimately connected with the discussion of neutral beam emergence from reconnection sites by Martens (988) and by Litvinenko & Somov (995). Acknowledgements. MK would like to acknowledge the support and kind hospitality of the Department of Physics and Astronomy at Glasgow University. This work was supported by UK PPARC Research and Visitor Grants, by a British Council Collaboration UK-France Grant (JCB and AJC), and by Spinoza Foundation Visitor Funding from the University of Amsterdam (JCB). 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